Valuation of Loan Credit Default Swaps Correlated Prepayment and Default Risks with Stochastic Recovery Rate

In this paper, we establish an intensity based multi-factor model to value LCDS. The pricing model incorporates the modeling of default, prepayment and recovery risks. Using one factor model, negative correlation between the default and prepayment intensities and positive correlation between the default intensity and the loss given default are described. The interest rate and the house price are chosen as the relevant factors. Under these assumption, a Cauthy problem of PDE is derived, which has a closed-form solution. Based on the solution, numerical examples are provided.

rates should be non-negative, we model them as CIR related processes. In this way, we overcome the shortage of Zhen Wei's work, where the prepayment intensity might be negative. To deal with negative correlation, we describe that the negative correlation as an inverse proportion, which was first proposed by Ahn and Gao (1999) and developed by Wu, Jiang & Liang, (2011) and Qian, Jiang and Xu (2010). Then if one satisfies a CIR process, the other follows inverse CIR process. The problem induces a non-standard partial differential equation which is studied by Hurd & Kuzneton (2008). Common factors in our 2-factor model for prepayment, default and recovery rates are interest rate and inflection index, which all follow the CIR processes. Under these assumptions, pricing LCDS is modelled where an explicit solution is obtained.
The rest of this paper is organized as follows. In the next section, under some basic notations and assumptions, we establish a pricing model for LCDS. In the third section, we use two-factor model to describe the relationship of default, prepayment, and recovery rate. Then we derive PDE problems and obtain a closed-form of solution for the spread. From the solution, the numerical examples are given in the fourth section.

Notations and Assumptions
Consider a single-name LCDS contract. The contract is ended before expired time only if a default or a prepayment event happens. Consider a probability space ( , , )  P Y and set the following notations: If the first stopping time is during [t i , t i+1 ], the contract is unwound. In this case, by (2), the present value of the coupon accrual. Supposing the last coupon payment date before  is unwound. In this case, the present value of the coupon payment paid till i t is given by The buyer stops paying coupons once the prepayment or default occurs. Then the present value of the premium leg is given by On the other hand, using (3), the expected present value of the losses in case of default is: Hence, the value of LCDS spread at time t comes:

Default intensity, prepayment intensity and recovery rates
As mentioned before, on pricing LCDS, the default and the prepayment risks are negatively correlated. If stochastic recovery rate is involved, it is also negatively correlated with the default risk.
In reality, the factors that affect the default risk, prepayment risk and recovery are complicated. Researchers have shown that among the factors, the interest rate is the most important one. When interest rate decreases, borrowers may refinance their loans, which lead to higher prepayment rate. However, the situation is opposite to the default events, i.e. the probability of default decreases. Another important fact is inflation rate, which affects the default and recovery rates. The higher the inflation rate is, the higher recovery and lower default rate are. To meet these facts, we take two-factor model to describe the relationship of the default risk, prepayment risk and recovery. Interest rate t r and inflation rate t H can serve as the correlated factors of them as follows: It has been proved that t y is stable, and it cannot equal 0 or  if and only if 2 2k      ,  >0,  >0. Under this condition, the origin CIR process t H still remains mean reverting and nonnegative. The situation is the same to t r .

The solution under CIR process
Taking equations (5) into equation (4) Observing the equation, we can find the problem can be solved as long as the following three expectations are derived: 1.
Thus, we just need to solve the expectations in the following types: Thus the solution of (12) is:   Figure 1 shows the relationship of LCDS spread with 0  Figure 1 and Figure2 that the effect of the inflection plays a more important role than that of the interest. Figure 3 indicates that the trend of LCDS spread with stochastic recovery rate is quite different from the one with the constant recovery rate. It suggests that the stochastic property of recovery rate gives the LCDS Spread volatility and the LCDS spread becomes more varying. We also find that the LCDS spread has a hump type shape which is similar to the one of many stochastic interest products.

Conclusion
In this paper, pricing LCDS is considered. We develop an intensity based multi-factor model, which incorporates the joint modeling of default, prepayment and recovery risks. We identified two correlations as risk factors that may explain the correlation among default intensity, prepayment intensities and recovery rates. The two common factors are assumed to be interest and inflection rates, which follow CIR process. This structure endure that the rates are non-negative and satisfy positive or negative correlations. A closed-form solution is obtained. Some numerical calculation examples are given, for which we get more direct view of the relationship among the parameters. The graph of the model shows that the inflection impacts the LCDS spread more than the interest does.